{{short description|Curve generated by rolling a circle inside another circle with 4x or (4/3)x the radius}} {{Distinguish|Asteroid}} thumb|Astroid thumb|The hypocycloid construction of the astroid. [[File:Astroid created with Elipses with a plus b const.svg|thumb|Astroid {{math|1= ''x''{{sup|{{frac|2|3}}}} + ''y''{{sup|{{frac|2|3}}}} = ''r''{{sup|{{frac|2|3}}}}}} as the common envelope of a family of ellipses of equation {{math|1= ({{frac|''x''|''a''}}){{sup|2}} + ({{frac|''y''|''b''}}){{sup|2}} = ''r''{{sup|2}}}}, where {{math|1= ''a'' + ''b'' = 1}}.]] [[File:sliding_ladder_in_astroid.svg|thumb|link={{filepath:sliding_ladder_in_astroid.svg}}|The envelope of a ladder (coloured lines in the top-right quadrant) sliding down a vertical wall, and its reflections (other quadrants) is an astroid. The midpoints trace out a circle while other points trace out ellipses similar to the previous figure. [{{filepath:sliding_ladder_in_astroid.svg}} {{nowrap|In the SVG file,}}] hover over a ladder to highlight it.]] thumb|right|Astroid as an evolute of ellipse
In mathematics, an '''astroid''' is a particular type of roulette curve: a hypocycloid with four cusps. Specifically, it is the locus of a point on a circle as it rolls inside a fixed circle with four times the radius.<ref>Yates</ref> By double generation, it is also the locus of a point on a circle as it rolls inside a fixed circle with 4/3 times the radius. It can also be defined as the envelope of a line segment of fixed length that moves while keeping an end point on each of the axes. It is therefore the envelope of the moving bar in the Trammel of Archimedes.
Its modern name comes from the Greek word for "star". It was proposed, originally in the form of "Astrois", by Joseph Johann von Littrow in 1838.<ref>{{cite book|author=J. J. v. Littrow|title=Kurze Anleitung zur gesammten Mathematik|chapter=§99. Die Astrois|year=1838|location=Wien|pages=299|chapter-url=https://books.google.com/books?id=AERmAAAAcAAJ&pg=PA299}}</ref><ref>{{cite book|author=Loria, Gino|title=Spezielle algebraische und transscendente ebene kurven. Theorie und Geschichte|url=https://archive.org/details/speziellealgebr00lorigoog|year=1902|location=Leipzig|pages=[https://archive.org/details/speziellealgebr00lorigoog/page/n250 224]}}</ref> The curve had a variety of names, including '''tetracuspid''' (still used), '''cubocycloid''', and '''paracycle'''. It is nearly identical in form to the evolute of an ellipse.
==Equations== If the radius of the fixed circle is ''a'' then the equation is given by<ref>Yates, for section</ref> <math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math> This means that an astroid is also a superellipse.
Parametric equations are <math display="block"> \begin{align} x = a\cos^3 t &= \frac{a}{4} \left( 3\cos \left(t\right) + \cos \left(3t\right)\right), \\[2ex] y = a\sin^3 t &= \frac{a}{4} \left( 3\sin \left(t\right) - \sin \left(3t\right) \right). \end{align} </math>
The pedal equation with respect to the origin is <math display="block">r^2 = a^2 - 3p^2,</math>
the Whewell equation is <math display="block">s = {3a \over 4} \cos 2\varphi,</math> and the Cesàro equation is <math display="block">R^2 + 4s^2 = \frac{9a^2}{4}.</math>
The polar equation is<ref>{{MathWorld | urlname=Astroid | title=Astroid}}</ref> <math display="block">r = \frac{a}{\left(\cos^{2/3}\theta + \sin^{2/3}\theta\right)^{3/2}}.</math>
The astroid is a real locus of a plane algebraic curve of genus zero. It has the equation<ref>A derivation of this equation is given on p. 3 of http://xahlee.info/SpecialPlaneCurves_dir/Astroid_dir/astroid.pdf</ref> <math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 a^2 x^2 y^2 = 0. </math>
The astroid is, therefore, a real algebraic curve of degree six.
==Derivation of the polynomial equation== The polynomial equation may be derived from Leibniz's equation by elementary algebra: <math display="block">x^{2/3} + y^{2/3} = a^{2/3}. </math>
Cube both sides: <math display="block">\begin{align} x^{6/3} + 3x^{4/3}y^{2/3} + 3x^{2/3}y^{4/3} + y^{6/3} &= a^{6/3} \\[1.5ex] x^2 + 3x^{2/3}y^{2/3} \left(x^{2/3} + y^{2/3}\right) + y^2 &= a^2 \\[1ex] x^2 + y^2 - a^2 &= -3x^{2/3}y^{2/3} \left(x^{2/3} + y^{2/3}\right) \end{align}</math>
Cube both sides again: <math display="block">\left(x^2 + y^2 - a^2\right)^3 = -27 x^2 y^2 \left(x^{2/3} + y^{2/3}\right)^3</math>
But since: <math display="block">x^{2/3} + y^{2/3} = a^{2/3} \,</math>
It follows that <math display="block">\left(x^{2/3} + y^{2/3}\right)^3 = a^2.</math>
Therefore: <math display="block">\left(x^2 + y^2 - a^2\right)^3 = -27 x^2 y^2 a^2</math> or <math display="block">\left(x^2 + y^2 - a^2\right)^3 + 27 x^2 y^2 a^2 = 0. </math>
==Metric properties== ;Area enclosed<ref>Yates, for section</ref> :<math>\frac{3}{8} \pi a^2</math> ;Length of curve :<math>6a</math> ;Volume of the surface of revolution of the enclose area about the ''x''-axis. :<math>\frac{32}{105}\pi a^3</math> ;Area of surface of revolution about the ''x''-axis :<math>\frac{12}{5}\pi a^2</math> ;Radius of the outscribed circle: <math>a</math> ;Radius of the inscribed circle: <math>a/2</math>
==Properties== The astroid has four cusp singularities in the real plane, the points on the star. It has two more complex cusp singularities at infinity, and four complex double points, for a total of ten singularities.
The dual curve to the astroid is the cruciform curve with equation <math display="inline"> x^2 y^2 = x^2 + y^2.</math> The evolute of an astroid is an astroid twice as large.
The astroid has only one tangent line in each oriented direction, making it an example of a hedgehog.<ref>{{cite journal | last1 = Nishimura | first1 = Takashi | last2 = Sakemi | first2 = Yu | doi = 10.14492/hokmj/1319595861 | issue = 3 | journal = Hokkaido Mathematical Journal | mr = 2883496 | pages = 361–373 | title = View from inside | volume = 40 | year = 2011| doi-access = free }}</ref>
==See also== * Cardioid – an epicycloid with one cusp * Deltoid – a hypocycloid with three cusps * Nephroid – an epicycloid with two cusps * Spirograph * Stoner–Wohlfarth astroid – a use of this curve in magnetics
==References== {{Reflist}} * {{Cite book | author=J. Dennis Lawrence | title=A catalog of special plane curves | url=https://archive.org/details/catalogspecialpl00lawr | url-access=limited | publisher=Dover Publications | year=1972 | isbn=0-486-60288-5 | pages=[https://archive.org/details/catalogspecialpl00lawr/page/n20 4]–5,34–35,173–174 }} * {{Cite book | author = Wells D | year = 1991 | title = The Penguin Dictionary of Curious and Interesting Geometry | publisher = Penguin Books | location = New York | isbn = 0-14-011813-6 | pages = 10–11}} * {{Cite book | author=R.C. Yates | title=A Handbook on Curves and Their Properties | location=Ann Arbor, MI | publisher=J. W. Edwards | pages=1 ff|chapter=Astroid| year=1952 }}
== External links == {{commons category|Astroid}} * {{springer|title=Astroid|id=p/a013540}} * [http://www-history.mcs.st-andrews.ac.uk/history/Curves/Astroid.html "Astroid" at The MacTutor History of Mathematics archive] * [http://www.mathcurve.com/courbes2d.gb/astroid/astroid.shtml "Astroid" at The Encyclopedia of Remarkable Mathematical Forms] * [https://web.archive.org/web/20170824133613/http://www.2dcurves.com/roulette/roulettea.html Article on 2dcurves.com] * [http://xahlee.org/SpecialPlaneCurves_dir/Astroid_dir/astroid.html Visual Dictionary Of Special Plane Curves, Xah Lee] * [http://demonstrations.wolfram.com/BarsOfAnAstroid/ Bars of an Astroid] by Sándor Kabai, The Wolfram Demonstrations Project.
Category:Sextic curves Category:Roulettes (curve)